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LIMITS AND CONTINUITY

THE CONCEPT OF LIMITS AND CONTINUITY

THE MEANING OF  Limit _xèa

Let x be a variable and a be a constant. Since x is a variable, its value can be so changed that its value comes nearer and nearer to a. Then we say that x approaches a and express it by the notation _xèa.

THE LIMIT OF A FUNCTION

The limit of a function for a given value of the variable is that constant to which the function continually approaches as the variable approaches the given value such that the difference between the constant and the function may be made as small as we please by making the variable approach sufficiently near to its assigned value. 

The  left hand limit, written as limit _{x èa-} f(x), is the value that f(x) approaches as x approaches ‘a’ from the left, (i.e. x tends to a by taking values which are less than a).

The right hand limit, written as limit _xèa+ f(x), is the value that f(x) approaches as x approaches ‘a’ from the right, (i.e. x tends to a by taking values which are more than a).

If the left hand limit and the right hand limit at ‘a’ are equal, than we say that limit _xèa f(x) exists and

limit _xèa f(x) = limit _{x èa-} f(x) = limit _{x èa+} f(x).

WORKING RULE FOR FINDING THE LEFT AND RIGHT HAND LIMITS

1. To find the left hand limit

• Put x = a-h in f(x)

• Take the limit of f(a-h) as _hè0 i.e. lim_xèa- f(x) = lim_hè0 f(a-h)

2. To find the right hand limit

• Put x = a+h in f(x)

• Take the limit of f(a+h) as _hè0 i.e. lim_xèa- f(x) = lim_hè0 f(a+h)

In order to calculate the limit of a function, we use the following standard limits and the theorem given below:

SOME STANDARD LIMITS

1. limit _xèa sinx/x = 1

2. limit _xèa cos x = 1

3. limit _xè0 tanx /x =1

4. limit _xè0 ( a^x-1) /x = ln(a)

5. limit _xè0 (1+x)^1/x = e

6. limit _xèoo (1+1/x)^x = e

7. limit _xè0 (1+ mx)^1/x = e^m, where m is a constant

8. limit _xèoo (1 +m/x)^x = e^m, where m is a constant

9. limit _xè1 (x-1)/logx = 1

10. limit _xèoo e^x = P

In the following cases, the limit does not exist :

1. limit _xè0+1/x = + P;  limit _xè0- 1/x = - P.

2. limit _xè(p/2)+ tanx = - P;  limit _xè(p/2)- tanx = - P.

3. limit _xèn+ [x] = n;  limit _xèn- [x] = (n-1),  n E I.

4. limit _xè0+ |x|/x = +1;  limit _xè0- |x|/x = - P

SOME BASIC THEOREMS

If limit _xèa f(x) = l,  limit _xèa g(x) = m, then

i) limit _xèa( f(x) ) ± ( g(x) ) = l ± m

ii) limit _xèa( f(x) . g(x) ) = l.m

iii) limit _xèa{f(x)/g(x)} = l/m  provided m<>o.

Some Special Limits

Here we will discuss some important limits that everyone should be aware of. They are very useful in many branches of science.

,

for any a > 0.
.

.

,

for any number a.